# Properties

 Label 540.4.i Level $540$ Weight $4$ Character orbit 540.i Rep. character $\chi_{540}(181,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $24$ Newform subspaces $3$ Sturm bound $432$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$540 = 2^{2} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 540.i (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$9$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$3$$ Sturm bound: $$432$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(540, [\chi])$$.

Total New Old
Modular forms 684 24 660
Cusp forms 612 24 588
Eisenstein series 72 0 72

## Trace form

 $$24 q + 10 q^{5} + 12 q^{7} + O(q^{10})$$ $$24 q + 10 q^{5} + 12 q^{7} + 54 q^{11} - 24 q^{13} - 348 q^{17} - 60 q^{19} + 84 q^{23} - 300 q^{25} + 342 q^{29} - 60 q^{31} - 280 q^{35} + 336 q^{37} + 804 q^{41} - 258 q^{43} + 276 q^{47} - 630 q^{49} - 3432 q^{53} + 630 q^{59} + 822 q^{61} + 200 q^{65} - 186 q^{67} + 216 q^{71} - 1500 q^{73} + 96 q^{77} - 888 q^{79} - 228 q^{83} - 360 q^{85} - 828 q^{89} + 24 q^{91} - 520 q^{95} - 546 q^{97} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(540, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
540.4.i.a $2$ $31.861$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-5$$ $$7$$ $$q-5\zeta_{6}q^{5}+(7-7\zeta_{6})q^{7}+(-30+30\zeta_{6})q^{11}+\cdots$$
540.4.i.b $8$ $31.861$ 8.0.$$\cdots$$.1 None $$0$$ $$0$$ $$-20$$ $$13$$ $$q+(-5-5\beta _{4})q^{5}+(2\beta _{2}-3\beta _{4}-\beta _{5}+\cdots)q^{7}+\cdots$$
540.4.i.c $14$ $31.861$ $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ None $$0$$ $$0$$ $$35$$ $$-8$$ $$q+(5-5\beta _{7})q^{5}+(-\beta _{7}-\beta _{9})q^{7}+(\beta _{1}+\cdots)q^{11}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(540, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(540, [\chi]) \cong$$ $$S_{4}^{\mathrm{new}}(9, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(18, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(27, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(36, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(54, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(90, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(108, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(135, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(180, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(270, [\chi])$$$$^{\oplus 2}$$